Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15)(7,16,17,20)(8,18,19,12)(9,14,10,13)(21,22,26,29)(23,31,34,35)(24,30,25,28)(27,33,36,32);(2,4)(5,7,11,17)(6,8,15,19)(9,12,10,18)(13,16,14,20)(21,23,26,34)(22,27,29,36)(24,33,25,32)(28,35,30,31);(5,8)(6,12)(7,14)(9,16)(10,20)(11,19)(13,17)(15,18)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(5,9,11,10)(6,13,15,14)(7,12,17,18)(8,16,19,20)(22,29)(27,36)(28,30)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140622);